Decision-making Method for Low-rent Housing Construction Investment. Wei Zhang*, Liwen You

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1 5th International Conferene on Civil Enineerin and Transportation (ICCET 5) Deision-makin Method for Low-rent Housin Constrution Investment Wei Zhan*, Liwen You University of Siene and Tehnoloy Liaonin, Anshan Liaonin University of Siene and Tehnoloy Liaonin, Anshan Liaonin Keywords: Developers, Government, Low-rent housin, Multi-stae, Option ame Abstrat. The paper sets up a investment deision-makin model with the option ame thinkin. First, based on the option ame features low-rent housin projets have, the paper proposes some researh hypotheses on value, benefits, osts for the options. Seond, on the basis of the deision-makin path of a multi-stae binary tree model, respetively for the overnment and developers, the paper builds utility funtions, ost-input models and deision-makin models of the ontrat period. It indiates that by the use of rowth options in the shortened ontrat period, developers an still ensure the quality of low-rent housin onstrution. Finally, the validity of the model in the low-rent housin areement and the investment deision-makin proess is verified by the ase analysis. Introdution In China, the demand of low-rent housin is far reater than the supply, and to solve the outstandin problem is the fund. Today, the fund of low-rent housin onstrution is in overnment-led investment model. To hane this sinle mode, it must be introdued other investment entities in the onstrution of low-rent housin. By the introdution of this mehanism of ompetition, the overnment's finanial pressure an not only be redued, but also the tehnoloy and manaement level an be improved in the low-ost housin onstrution. Due to the lon investment reovery period and many unertain fators in the low-rent housin projets, the option ame method is used to establish the deision model of determinin the ontrat period, this paper puts forward the investment deision method of multi-stae low-rent housin onstrution, whih brins more flexibility for the deisions of the overnment and developers. The overnment has adopted multi-stae low-rent housin projets, namely the winnin-bid developer in the first stae an ontinue to invest in the next stae, whether to hoose suh investment opportunities and whether suh investment opportunities is of the value an be evaluated by rowth options. The winnin-bid developer determines whether to implement the rowth option based on the ompany s stratei oals, the net ash flow and expeted benefits rowth options brin and et. The both sides in the neotiation proess will measure the value of the rowth option, onsiderin whether it is worth implementin the rowth option. The overnment an investiate the winnin-bid developer whether there is suffiient tehnial apaity and manaement level durin the ontrat period, in order to determine whether or not to hoose the ompany to ontinue to invest in the seond stae. While the developer an base the ompany's stratei oals, onditions of investment opportunities and other fators to determine whether to ontinue the seond stae of the investment projet. The next staes proeed in turn. In short, the introdution of rowth options an not only avoid the risk flexibly, but also brin more hoies to both sides. Buildin the Model Basi Assumptions. The assumptions are as follows: Assumption one: The onstrution of low-rent housin projets is divided into two staes, the winnin-bid developer an perform the rowth option of the seond stae, whose value is expressed as. 5. The authors - Published by Atlantis Press 447

2 Assumption two: Durin the operation of the low-rent housin, P (i.e. the unit rent) should be set by the overnment, and the prie hanes over time is neliible. Developers obtain the profits by harin rent, and the differene between the rent and the market level is subsidized by the overnment. Assumption three: The annual averae of q (i.e. the amount of low-ost housin onstrution) is a I=p f(q)dq onstant. Based on the assumption two, the averae annual inome is expressed as. Here, f(q) is the distribution funtion of q, I is the averae annual inome. Assumption four: The late operational ost inludes manaement ost and annual review ost for qualifiations, and the manaement ost will inrease with the low-rent housin over time. The Manaement ost an be expressed as Cm = ktc (k>,>). Model Buildin. The overnment adopts the method that developers involve in the multi-stae investment of low-ost housin projets. The first stae is required for the neotiation about the ontrat period. For deision variables of the overnment and developers, we set the ontrat period as T, the onstrution ost as C, the eonomi life period of the low-rent housin as T. By Assumption one, the developer an hoose whether to ontinue to invest in the seond stae of the projet, based on the expeted benefits in the first stae of low-rent housin projet. Aordin to the binary tree method in the basi theory of real options, when the ontrat expires, the value of the first stae of the projet may appear as two states, + and - -. If the value of the projet is up to, the developer will stop to ontinue to invest the next stae. If the value of the projet is up to +, the developer will ontinue to invest in the seond stae of the projet, settin the investment ost as C, and the value of the seond stae may also appear as two states, + and -. Therefore, the value of the investment opportunities of low-ost housin projets is iven by: + + =max(, + C) () - - =max(, C) () The investment value of the low-rent housin projet ontainin rowth options, is iven by: + P + ( P) = C r + (3) P is risk neutral probability, r is the risk-free rate. In order to benefit the publi, the overnment in the neotiations will first determine the ontrat period T. Then, we an have the deision-makin model that deision variable T satisfies : max T [p T f(q)dq-c m]dt C (4) T C m dt C+ µ (5) s.t. [p f(q)dq-c ] Eq.4 is the utility funtion of the overnment to maximize the publi interest under the ontrat period. Eq.5 is the onstraint ondition whih the utility funtion of the overnment need to meet, namely that the expeted utility developers ahieve in the ontrat period of low-rent housin projets is not less than the maximum expeted utility ained by other projets. Where, the maximum expeted utility whih developers an ain from other projets, namely the opportunity utility, is expressed as μ. Aordin to the ontrat period T iven by the overnment and the profit maximization, Developers an determine C (i.e., onstrution ost), so the deision-makin model deision variables C satisfies: T max{ [p f(q)dq-c m] dt C+ } (6) Eq.6 is to asertain the onstrution ost C, based on the ontrat period T iven by the overnment and under the ondition of profit maximization. The T iven by the overnment is often less than the value obtained by the traditional method, even if there are some onstraints of rowth options, developers may also redue the input of onstrution ost. Solution and Disussion. In order that the benefits of the overnment and developers an ahieve the maximum, we an adopt the bakward indution to solve both the deision-makin models, that is the 448

3 first to determine C developers need to invest in aordane with the iven T, then the overnment will determine the appropriate T based on C. First, it is to determine C developers need to invest in aordane with the iven T. For I=p f(q)dq Assumption three and Assumption four, substitutin and Cm = ktc into Eq.6 ives the followin formula: T - max[ (I-ktC )dt C+ ] (7) Aordinly, Eq.7 is simplified: + max[it kt C C ] Based on the first order ondition of the optimization, Eq.8 an be derived: C = kt + (9) Then, the overnment will refer to the C developers pre-estimate, namely determinin the appropriate T based on C. Substitutin Eq.9 into Eq.4 and Eq.5, thus the deision model and the onstraint ondition for deision variable T an be written: T + max[i(t-t )- ( ) k T T tdt] - - T s.t. IT- ( ) k T tdt C+ µ () To solve Eq. under the onstrain ondition of Eq., we an use Larane multiplier method. Thus, we an onstrut a Larane funtion with Larane multiplier λ, the funtion is iven by: - - T L(T C, λ) = I(T-T C)- ( ) k T tdt - - T [I- ( ) k T tdt-c - - ] + λ + µ Respetively, T and λ in Eq. an be derived by solvin the first order partial derivative: L C T λ+ λ- = ( λ )I + [ - ] = ( + ) (3) L + = I C+ µ = λ (4) Solvin Eq.4, we obtain: + = C + µ = I I I (5) Then, we an analyze the relationships between onstrution ost C, ontrat period T and rowth option. Solvin the derivative of Eq.9, it results that: C ( k ) + + = T > T + (6) We an observe that C will inrease as T inreases.from Eq.5, we an know that the ontrat period T is relevant to the onstrution ost C, the averae annual inome I, the value of rowth option and the opportunity utility μ. Solvin the derivative of Eq.5, it results that: T = < C I (7) We an observe that T will inrease as dereases. Hene, the overnment will extend the ontrat period T to make up for the disadvantae of the rowth option. Solvin the derivative of Eq.5, it results that: = > µ I (8) We an observe that T will inrease as μ inreases. So the overnment an extend the ontrat period T to make ROI(i.e., Return on Investment) not less than other investments. Likewise, C will inrease as inreases under the iven T. So developers will inrease the investment due to the inreased rowth option. 449 (8) () ()

4 Case Analysis A overnment will prepare to build, units of low-rent housin, whih proeed in two staes. units will be built in the first stae, and 8 units will be built in the seond stae. Given that eah set low-rent house is 5 square meters, the ost per square meter is, yuan, and the entral finanial subsidy is, yuan per square meter. So developers are required to pay, yuan of mathin funds. We an onlude that the sale of investment in the first stae of the projet is 6 million yuan, and the sale of investment in the seond stae is 4 million yuan. A developer has won the bid and started to onstrut the first stae of the projet, its data are shown in the followin table. simbol C I μ K C P r data %. Table. The data in first stae(million) When the rowth option expires, the value in the first stae of the projet is: + - =6, =3. After investin the seond stae of the projet, + =, Aordin to Eq.,,3, we an obtain: + =8 Usin the Eq.5, we an write: ) Usin the Eq.9, we an obtain: + k 46 (million) - =87. =47 P + + ( P) = C = (million 3 + r ) + µ = C + = 8.8 (year I I I C = T = From the alulation results, we an be obtain that the onstrution ost of the developer in the nine-year ontrat period is 46 million yuan. Obviously, this result is lower than the ost the overnment requires, but the developer will not invest onstrution funds lower than the ost of the ontrat, beause if the developer redues the onstrution ost of investment in the first stae of the projet, then the overnment will deprive the developer of the riht of its ontinued onstrution of the seond stae. Conlusion In the onstrution of low-rent housin projets, the neotiations of ontrat areements are not only related to the profit distribution between the overnment and developers, but also affet the quality of the projet in the onstrution. Determinin the ontrat period is the key link of the neotiations of ontrat areements. Therefore, based on the option ame features low-rent housin projets have, the paper builds utility funtions, ost-input models and deision-makin models of the ontrat period. The influenes iven by onstrution ost, ontrat period and other relevant deision-makin fators on the development areement are derived, with the quantitative relationships between the ontrat period and its influenin fators. It indiates that by the use of rowth options in the shortened ontrat period, developers an still ensure the quality of low-rent housin onstrution. Referenes [] Guoxin Zhan, Xiulin Gao, Yinluo Wan, An asymmetri duopoly investment deision-makin model based on differene of option, Systems Enineerin-Theory & Pratie 35(3):75-76,5. [] Guanxu Wu, Xu Ren, The Game Deision Analysis of Low-rent Housin PPP Mode, Journal of Enineerin Manaement 6(3):94-98,. [3] Wei Li & Daijin Xi, Option Game Analysis on Symmetri Enterprise R&D Projet under Unertain Conditions, alue Enineerin ():-,. 45

5 [4] Yulin Liao, Qianlin Hon, The Timin Seletin of Investment Based on Option Game, The Theory and Pratie of Finane and Eonomis 3(7):3-34,. 45

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